-
@@ -219,7 +223,7 @@ MathJax.Hub.Config({
-Random forests provide an improvement over bagged trees by way of a -small tweak that decorrelates the trees. +
-As in bagging, we build a -number of decision trees on bootstrapped training samples. But when -building these decision trees, each time a split in a tree is -considered, a random sample of \( m \) predictors is chosen as split -candidates from the full set of \( p \) predictors. The split is allowed to -use only one of those \( m \) predictors. - -
-A fresh sample of \( m \) predictors is -taken at each split, and typically we choose - -$$ -m\approx \sqrt{p}. -$$ - -
-In building a random forest, at -each split in the tree, the algorithm is not even allowed to consider -a majority of the available predictors. - -
-The reason for this is rather clever. Suppose that there is one very -strong predictor in the data set, along with a number of other -moderately strong predictors. Then in the collection of bagged -variable importance random forest trees, most or all of the trees will -use this strong predictor in the top split. Consequently, all of the -bagged trees will look quite similar to each other. Hence the -predictions from the bagged trees will be highly correlated. -Unfortunately, averaging many highly correlated quantities does not -lead to as large of a reduction in variance as averaging many -uncorrelated quantities. In particular, this means that bagging will -not lead to a substantial reduction in variance over a single tree in -this setting. +Geron's chapter 7. See also lecture from STK-IN4300, lecture 7. Chapter 9.2 of Hastie et al contains also a good discussion.
@@ -234,7 +206,7 @@ this setting.
-We will grow of forest of say \( B \) trees. - -
diff --git a/doc/pub/week45/html/._week45-bs003.html b/doc/pub/week45/html/._week45-bs003.html index 515520c4c..632efc707 100644 --- a/doc/pub/week45/html/._week45-bs003.html +++ b/doc/pub/week45/html/._week45-bs003.html @@ -41,64 +41,66 @@ Automatically generated HTML file from DocOnce source @@ -136,31 +138,33 @@ MathJax.Hub.Config({
+Random forests provide an improvement over bagged trees by way of a +small tweak that decorrelates the trees. - -
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.datasets import load_breast_cancer
-from sklearn.svm import SVC
-from sklearn.linear_model import LogisticRegression
-from sklearn.tree import DecisionTreeClassifier
+
+As in bagging, we build a
+number of decision trees on bootstrapped training samples. But when
+building these decision trees, each time a split in a tree is
+considered, a random sample of \( m \) predictors is chosen as split
+candidates from the full set of \( p \) predictors. The split is allowed to
+use only one of those \( m \) predictors.
-# Load the data
-cancer = load_breast_cancer()
+
+A fresh sample of \( m \) predictors is
+taken at each split, and typically we choose
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-# Logistic Regression
-logreg = LogisticRegression(solver='lbfgs')
-logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
-# Support vector machine
-svm = SVC(gamma='auto', C=100)
-svm.fit(X_train, y_train)
-print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
-# Decision Trees
-deep_tree_clf = DecisionTreeClassifier(max_depth=None)
-deep_tree_clf.fit(X_train, y_train)
-print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Logistic Regression
-logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-# Support Vector Machine
-svm.fit(X_train_scaled, y_train)
-print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-# Decision Trees
-deep_tree_clf.fit(X_train_scaled, y_train)
-print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
+$$
+m\approx \sqrt{p}.
+$$
+
+In building a random forest, at
+each split in the tree, the algorithm is not even allowed to consider
+a majority of the available predictors.
-from sklearn.ensemble import RandomForestClassifier
-from sklearn.preprocessing import LabelEncoder
-from sklearn.model_selection import cross_validate
-# Data set not specificied
-#Instantiate the model with 500 trees and entropy as splitting criteria
-Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy")
-Random_Forest_model.fit(X_train_scaled, y_train)
-#Cross validation
-accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
+
+The reason for this is rather clever. Suppose that there is one very
+strong predictor in the data set, along with a number of other
+moderately strong predictors. Then in the collection of bagged
+variable importance random forest trees, most or all of the trees will
+use this strong predictor in the top split. Consequently, all of the
+bagged trees will look quite similar to each other. Hence the
+predictions from the bagged trees will be highly correlated.
+Unfortunately, averaging many highly correlated quantities does not
+lead to as large of a reduction in variance as averaging many
+uncorrelated quantities. In particular, this means that bagging will
+not lead to a substantial reduction in variance over a single tree in
+this setting.
-
-import scikitplot as skplt
-y_pred = Random_Forest_model.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-plt.show()
-y_probas = Random_Forest_model.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-plt.show()
-
@@ -265,7 +240,7 @@ plt.show()
+
bag_clf = BaggingClassifier(
- DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
- n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
-+We will grow of forest of say \( B \) trees. + +
bag_clf.fit(X_train, y_train)
-y_pred = bag_clf.predict(X_test)
-from sklearn.ensemble import RandomForestClassifier
-rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
-rnd_clf.fit(X_train, y_train)
-y_pred_rf = rnd_clf.predict(X_test)
-np.sum(y_pred == y_pred_rf) / len(y_pred)
-
diff --git a/doc/pub/week45/html/._week45-bs005.html b/doc/pub/week45/html/._week45-bs005.html index 76e049c98..00fe570d3 100644 --- a/doc/pub/week45/html/._week45-bs005.html +++ b/doc/pub/week45/html/._week45-bs005.html @@ -41,64 +41,66 @@ Automatically generated HTML file from DocOnce source @@ -136,31 +138,33 @@ MathJax.Hub.Config({
-The basic idea is to combine weak classifiers in order to create a good -classifier. With a weak classifier we often intend a classifier which -produces results which are only slightly better than we would get by -random guesses. -
-This is done by applying in an iterative way a weak (or a standard -classifier like decision trees) to modify the data. In each iteration -we emphasize those observations which are misclassified by weighting -them with a factor. + +
import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import train_test_split
+from sklearn.datasets import load_breast_cancer
+from sklearn.svm import SVC
+from sklearn.linear_model import LogisticRegression
+from sklearn.tree import DecisionTreeClassifier
+# Load the data
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+# Logistic Regression
+logreg = LogisticRegression(solver='lbfgs')
+logreg.fit(X_train, y_train)
+print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
+# Support vector machine
+svm = SVC(gamma='auto', C=100)
+svm.fit(X_train, y_train)
+print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
+# Decision Trees
+deep_tree_clf = DecisionTreeClassifier(max_depth=None)
+deep_tree_clf.fit(X_train, y_train)
+print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
+#now scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Logistic Regression
+logreg.fit(X_train_scaled, y_train)
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+# Support Vector Machine
+svm.fit(X_train_scaled, y_train)
+print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+# Decision Trees
+deep_tree_clf.fit(X_train_scaled, y_train)
+print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
+
+
+from sklearn.ensemble import RandomForestClassifier
+from sklearn.preprocessing import LabelEncoder
+from sklearn.model_selection import cross_validate
+# Data set not specificied
+#Instantiate the model with 500 trees and entropy as splitting criteria
+Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy")
+Random_Forest_model.fit(X_train_scaled, y_train)
+#Cross validation
+accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
+
+
+import scikitplot as skplt
+y_pred = Random_Forest_model.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+plt.show()
+y_probas = Random_Forest_model.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test, y_probas)
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test, y_probas)
+plt.show()
+
@@ -211,7 +271,7 @@ them with a factor.
-Boosting is a way of fitting an additive expansion in a set of -elementary basis functions like for example some simple polynomials. -Assume for example that we have a function -$$ -f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), -$$ + +
bag_clf = BaggingClassifier(
+ DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
+ n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
+-where \( \beta_m \) are the expansion parameters to be determined in a -minimization process and \( b(x;\gamma_m) \) are some simple functions of -the multivariable parameter \( x \) which is characterized by the -parameters \( \gamma_m \). - -
-As an example, consider the Sigmoid function we used in logistic -regression. In that case, we can translate the function -\( b(x;\gamma_m) \) into the Sigmoid function - -$$ -\sigma(t) = \frac{1}{1+\exp{(-t)}}, -$$ - -
-where \( t=\gamma_0+\gamma_1 x \) and the parameters \( \gamma_0 \) and -\( \gamma_1 \) were determined by the Logistic Regression fitting -algorithm. - -
-As another example, consider the cost function we defined for linear regression -$$ -C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. -$$ - -
-In this case the function \( f(x) \) was replaced by the design matrix -\( \boldsymbol{X} \) and the unknown linear regression parameters \( \boldsymbol{\beta} \), -that is \( \boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta} \). In linear regression we can -simply invert a matrix and obtain the parameters \( \beta \) by - -$$ -\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. -$$ - -
-In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \( \beta_m \) and \( \gamma_m \). + +
bag_clf.fit(X_train, y_train)
+y_pred = bag_clf.predict(X_test)
+from sklearn.ensemble import RandomForestClassifier
+rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
+rnd_clf.fit(X_train, y_train)
+y_pred_rf = rnd_clf.predict(X_test)
+np.sum(y_pred == y_pred_rf) / len(y_pred)
+
@@ -247,7 +221,7 @@ In iterative fitting or additive modeling, we minimize the cost function with re
-The way we proceed is as follows (here we specialize to the squared-error cost function) +The basic idea is to combine weak classifiers in order to create a good +classifier. With a weak classifier we often intend a classifier which +produces results which are only slightly better than we would get by +random guesses. -
+This is done by applying in an iterative way a weak (or a standard +classifier like decision trees) to modify the data. In each iteration +we emphasize those observations which are misclassified by weighting +them with a factor.
@@ -221,7 +217,7 @@ at the internal nodes, and the predictions at the terminal nodes.
-To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function. - -
-For simplicity we assume also that our functions \( b(x;\gamma)=1+\gamma x \). - -
-This means that for every iteration \( m \), we need to optimize - +Boosting is a way of fitting an additive expansion in a set of +elementary basis functions like for example some simple polynomials. +Assume for example that we have a function $$ -(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2. +f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), $$
-We start our iteration by simply setting \( f_0(x)=0 \). -Taking the derivatives with respect to \( \beta \) and \( \gamma \) we obtain -$$ -\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0, -$$ +where \( \beta_m \) are the expansion parameters to be determined in a +minimization process and \( b(x;\gamma_m) \) are some simple functions of +the multivariable parameter \( x \) which is characterized by the +parameters \( \gamma_m \). -and -$$ -\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0. -$$ +
+As an example, consider the Sigmoid function we used in logistic +regression. In that case, we can translate the function +\( b(x;\gamma_m) \) into the Sigmoid function -We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector) $$ -\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, -$$ - -which gives us \( \beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w}) \). Similarly we have -$$ -\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0, +\sigma(t) = \frac{1}{1+\exp{(-t)}}, $$
-which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting -for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically. +where \( t=\gamma_0+\gamma_1 x \) and the parameters \( \gamma_0 \) and +\( \gamma_1 \) were determined by the Logistic Regression fitting +algorithm.
-The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma_1 \) leading to the new expression for \( f_1(x) \) as -\( f_1(x) = \beta_1(1+\gamma_1x) \). Doing this \( M \) times results in our final estimate for the function \( f \). +As another example, consider the cost function we defined for linear regression +$$ +C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. +$$ + +
+In this case the function \( f(x) \) was replaced by the design matrix +\( \boldsymbol{X} \) and the unknown linear regression parameters \( \boldsymbol{\beta} \), +that is \( \boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta} \). In linear regression we can +simply invert a matrix and obtain the parameters \( \beta \) by + +$$ +\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +$$ + +
+In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \( \beta_m \) and \( \gamma_m \).
@@ -245,7 +253,7 @@ The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma
-Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of -observations. We define a classification function \( G(x) \) which produces a prediction taking one or the other of the two values -\( \{-1,1\} \). +The way we proceed is as follows (here we specialize to the squared-error cost function) -
-The error rate of the training sample is then +
-The iterative procedure starts with defining a weak classifier whose -error rate is barely better than random guessing. The iterative -procedure in boosting is to sequentially apply a weak -classification algorithm to repeatedly modified versions of the data -producing a sequence of weak classifiers \( G_m(x) \). + -
-Here we will express our function \( f(x) \) in terms of \( G(x) \). That is -$$ -f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), -$$ - -will be a function of -$$ -G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x). -$$ +We could use any of the algorithms we have discussed till now. If we +use trees, \( \gamma \) parameterizes the split variables and split points +at the internal nodes, and the predictions at the terminal nodes.
@@ -233,7 +227,7 @@ $$
-In our iterative procedure we define thus +To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function. + +
+For simplicity we assume also that our functions \( b(x;\gamma)=1+\gamma x \). + +
+This means that for every iteration \( m \), we need to optimize + $$ -f_m(x) = f_{m-1}(x)+\beta_mG_m(x). +(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2. $$
-The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the -exponential cost/loss function defined as +We start our iteration by simply setting \( f_0(x)=0 \). +Taking the derivatives with respect to \( \beta \) and \( \gamma \) we obtain $$ -C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}. +\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0, +$$ + +and +$$ +\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0. +$$ + +We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector) +$$ +\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, +$$ + +which gives us \( \beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w}) \). Similarly we have +$$ +\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0, $$
-We optimize \( \beta \) and \( G \) for each value of \( m=1:M \) as we did in the regression case. -This is normally done in two steps. Let us however first rewrite the cost function as +which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting +for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically. -$$ -C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))}, -$$ - -where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \). +
+The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma_1 \) leading to the new expression for \( f_1(x) \) as +\( f_1(x) = \beta_1(1+\gamma_1x) \). Doing this \( M \) times results in our final estimate for the function \( f \).
@@ -227,7 +251,7 @@ where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
-First, for any \( \beta > 0 \), we optimize \( G \) by setting -$$ -G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)), -$$ - -which is the classifier that minimizes the weighted error rate in predicting \( y \). +Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of +observations. We define a classification function \( G(x) \) which produces a prediction taking one or the other of the two values +\( \{-1,1\} \).
-We can do this by rewriting +The error rate of the training sample is then + $$ -\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m, +\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)). $$ -which can be rewritten as +
+The iterative procedure starts with defining a weak classifier whose +error rate is barely better than random guessing. The iterative +procedure in boosting is to sequentially apply a weak +classification algorithm to repeatedly modified versions of the data +producing a sequence of weak classifiers \( G_m(x) \). + +
+Here we will express our function \( f(x) \) in terms of \( G(x) \). That is $$ -(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0, +f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), $$ -which leads to +will be a function of $$ -\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}}, -$$ - -where we have redefined the error as -$$ -\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m}, -$$ - -which leads to an update of -$$ -f_m(x) = f_{m-1}(x) +\beta_m G_m(x). -$$ - -This leads to the new weights -$$ -w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))} +G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x). $$
@@ -243,7 +238,7 @@ $$
-The algorithm here is rather straightforward. Assume that our weak -classifier is a decision tree and we consider a binary set of outputs -with \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of -observations. Our design matrix is given in terms of the -feature/predictor vectors -\( \boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}] \). Finally, we define also a -classifier determined by our data via a function \( G(x) \). This function tells us how well we are able to classify our outputs/targets \( \boldsymbol{y} \). +In our iterative procedure we define thus +$$ +f_m(x) = f_{m-1}(x)+\beta_mG_m(x). +$$
-We have already defined the misclassification error \( \mathrm{err} \) as +The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the +exponential cost/loss function defined as $$ -\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)), +C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}. $$ -where the function \( I() \) is one if we misclassify and zero if we classify correctly. +
+We optimize \( \beta \) and \( G \) for each value of \( m=1:M \) as we did in the regression case. +This is normally done in two steps. Let us however first rewrite the cost function as + +$$ +C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))}, +$$ + +where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
@@ -221,7 +231,7 @@ where the function \( I() \) is one if we misclassify and zero if we classify co
-With the above definitions we are now ready to set up the algorithm for AdaBoost. -The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases. - -
+We can do this by rewriting +$$ +\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m, +$$ -
@@ -239,7 +247,7 @@ observations that are missed in the previous iterations.
-Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here. +The algorithm here is rather straightforward. Assume that our weak +classifier is a decision tree and we consider a binary set of outputs +with \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of +observations. Our design matrix is given in terms of the +feature/predictor vectors +\( \boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}] \). Finally, we define also a +classifier determined by our data via a function \( G(x) \). This function tells us how well we are able to classify our outputs/targets \( \boldsymbol{y} \).
+We have already defined the misclassification error \( \mathrm{err} \) as +$$ +\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)), +$$ - -
from sklearn.ensemble import AdaBoostClassifier
+where the function \( I() \) is one if we misclassify and zero if we classify correctly.
-ada_clf = AdaBoostClassifier(
- DecisionTreeClassifier(max_depth=1), n_estimators=200,
- algorithm="SAMME.R", learning_rate=0.5, random_state=42)
-ada_clf.fit(X_train, y_train)
-
-from sklearn.ensemble import AdaBoostClassifier
-
-ada_clf = AdaBoostClassifier(
- DecisionTreeClassifier(max_depth=1), n_estimators=200,
- algorithm="SAMME.R", learning_rate=0.5, random_state=42)
-ada_clf.fit(X_train_scaled, y_train)
-y_pred = ada_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-plt.show()
-y_probas = ada_clf.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-plt.show()
-
@@ -232,7 +225,7 @@ plt.show()
-Here we present Drucker's AdaBoost tailored for regression. - -
-In bagging, each training example is equally likely to be -picked. In boosting, the probability of a particular -example being in the training set of a particular machine -depends on the performance of the prior machines on -that example. The following is a modification of -Adaboost by Drucker. - -
-Start by selecting a set of training data \( n \) and assign to each entry a weight \( w_i=1 \) for \( i=1,2,\dots,n \). As we have done earlier, we could pick say \( 80\% \) of the data set for training. The algorithm runs as follows: +With the above definitions we are now ready to set up the algorithm for AdaBoost. +The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
@@ -227,7 +243,7 @@ Start by selecting a set of training data \( n \) and assign to each entry a wei
-Gradient boosting is again a similar technique to Adaptive boosting, -it combines so-called weak classifiers or regressors into a strong -method via a series of iterations. +Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here.
-In order to understand the method, let us illustrate its basics by -bringing back the essential steps in linear regression, where our cost -function was the least squares function. + +
from sklearn.ensemble import AdaBoostClassifier
+
+ada_clf = AdaBoostClassifier(
+ DecisionTreeClassifier(max_depth=1), n_estimators=200,
+ algorithm="SAMME.R", learning_rate=0.5, random_state=42)
+ada_clf.fit(X_train, y_train)
+
+from sklearn.ensemble import AdaBoostClassifier
+
+ada_clf = AdaBoostClassifier(
+ DecisionTreeClassifier(max_depth=1), n_estimators=200,
+ algorithm="SAMME.R", learning_rate=0.5, random_state=42)
+ada_clf.fit(X_train_scaled, y_train)
+y_pred = ada_clf.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+plt.show()
+y_probas = ada_clf.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test, y_probas)
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test, y_probas)
+plt.show()
+
@@ -213,6 +235,8 @@ function was the least squares function.
-We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize -This means that for every iteration, we need to optimize - -$$ -(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. -$$ +Here we present Drucker's AdaBoost tailored for regression.
-We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as -$$ -f_M(x) = \sum_{m=0}^M h_m(x). -$$ +In bagging, each training example is equally likely to be +picked. In boosting, the probability of a particular +example being in the training set of a particular machine +depends on the performance of the prior machines on +that example. The following is a modification of +Adaboost by Drucker.
-In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as -$$ -g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. -$$ +Start by selecting a set of training data \( n \) and assign to each entry a weight \( w_i=1 \) for \( i=1,2,\dots,n \). As we have done earlier, we could pick say \( 80\% \) of the data set for training. The algorithm runs as follows: -
-With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that -the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \). +
-Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have -$$ -(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. -$$ +\( L_i\in [0,1] \).
@@ -232,6 +229,9 @@ $$
-Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that -$$ -f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i. -$$ +Gradient boosting is again a similar technique to Adaptive boosting, +it combines so-called weak classifiers or regressors into a strong +method via a series of iterations. -We can then proceed and compute -$$ -g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i, -$$ - -and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \( m=M \). We can modify the steepest descent method, or steepest boosting, by introducing what is called gradient boosting. +
+In order to understand the method, let us illustrate its basics by +bringing back the essential steps in linear regression, where our cost +function was the least squares function.
@@ -214,6 +215,8 @@ and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \(
-Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function +We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize +This means that for every iteration, we need to optimize + $$ -C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. +(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. $$
-The way we proceed in an iterative fashion is to +We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as +$$ +f_M(x) = \sum_{m=0}^M h_m(x). +$$ -
+In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as +$$ +g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. +$$ -
+With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that +the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \). -
+Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have +$$ +(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. +$$ +
diff --git a/doc/pub/week45/html/._week45-bs020.html b/doc/pub/week45/html/._week45-bs020.html index 56c7c37d7..280b59923 100644 --- a/doc/pub/week45/html/._week45-bs020.html +++ b/doc/pub/week45/html/._week45-bs020.html @@ -41,64 +41,66 @@ Automatically generated HTML file from DocOnce source @@ -136,31 +138,33 @@ MathJax.Hub.Config({
-We discuss here the difference between the steepest descent approach and gradient boosting by repeating our simple regression example above. +Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that +$$ +f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i. +$$ + +We can then proceed and compute +$$ +g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i, +$$ + +and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \( m=M \). We can modify the steepest descent method, or steepest boosting, by introducing what is called gradient boosting.
@@ -202,6 +216,8 @@ We discuss here the difference between the steepest descent approach and gradien
+Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function +$$ +C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. +$$ - -
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.ensemble import GradientBoostingRegressor
-from sklearn.preprocessing import StandardScaler
-import scikitplot as skplt
-from sklearn.metrics import mean_squared_error
-
-n = 100
-maxdegree = 6
-
-# Make data set.
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-
-error = np.zeros(maxdegree)
-bias = np.zeros(maxdegree)
-variance = np.zeros(maxdegree)
-polydegree = np.zeros(maxdegree)
-X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-for degree in range(1,maxdegree):
- model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0)
- model.fit(X_train_scaled,y_train)
- y_pred = model.predict(X_test_scaled)
- polydegree[degree] = degree
- error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
- bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
- variance[degree] = np.mean( np.var(y_pred) )
- print('Max depth:', degree)
- print('Error:', error[degree])
- print('Bias^2:', bias[degree])
- print('Var:', variance[degree])
- print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
-
-plt.xlim(1,maxdegree-1)
-plt.plot(polydegree, error, label='Error')
-plt.plot(polydegree, bias, label='bias')
-plt.plot(polydegree, variance, label='Variance')
-plt.legend()
-save_fig("gdregression")
-plt.show()
-+The way we proceed in an iterative fashion is to + +
diff --git a/doc/pub/week45/html/._week45-bs022.html b/doc/pub/week45/html/._week45-bs022.html index 51c974b7a..1dde295c0 100644 --- a/doc/pub/week45/html/._week45-bs022.html +++ b/doc/pub/week45/html/._week45-bs022.html @@ -41,64 +41,66 @@ Automatically generated HTML file from DocOnce source @@ -136,31 +138,33 @@ MathJax.Hub.Config({
+We discuss here the difference between the steepest descent approach and gradient boosting by repeating our simple regression example above. - -
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.datasets import load_breast_cancer
-import scikitplot as skplt
-from sklearn.ensemble import GradientBoostingClassifier
-from sklearn.model_selection import cross_validate
-
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0)
-gd_clf.fit(X_train_scaled, y_train)
-#Cross validation
-accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
-
-import scikitplot as skplt
-y_pred = gd_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-save_fig("gdclassiffierconfusion")
-plt.show()
-y_probas = gd_clf.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-save_fig("gdclassiffierroc")
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-save_fig("gdclassiffiercgain")
-plt.show()
-
@@ -240,6 +204,8 @@ plt.show()
-XGBoost or Extreme Gradient -Boosting, is an optimized distributed gradient boosting library -designed to be highly efficient, flexible and portable. It implements -machine learning algorithms under the Gradient Boosting -framework. XGBoost provides a parallel tree boosting that solve many -data science problems in a fast and accurate way. See the article by Chen and Guestrin. -
-The authors design and build a highly scalable end-to-end tree -boosting system. It has a theoretically justified weighted quantile -sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning. + +
import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import train_test_split
+from sklearn.ensemble import GradientBoostingRegressor
+from sklearn.preprocessing import StandardScaler
+import scikitplot as skplt
+from sklearn.metrics import mean_squared_error
-
-It is now the algorithm which wins essentially all ML competitions!!!
+n = 100
+maxdegree = 6
+# Make data set.
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+
+error = np.zeros(maxdegree)
+bias = np.zeros(maxdegree)
+variance = np.zeros(maxdegree)
+polydegree = np.zeros(maxdegree)
+X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+
+for degree in range(1,maxdegree):
+ model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0)
+ model.fit(X_train_scaled,y_train)
+ y_pred = model.predict(X_test_scaled)
+ polydegree[degree] = degree
+ error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
+ bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
+ variance[degree] = np.mean( np.var(y_pred) )
+ print('Max depth:', degree)
+ print('Error:', error[degree])
+ print('Bias^2:', bias[degree])
+ print('Var:', variance[degree])
+ print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+
+plt.xlim(1,maxdegree-1)
+plt.plot(polydegree, error, label='Error')
+plt.plot(polydegree, bias, label='bias')
+plt.plot(polydegree, variance, label='Variance')
+plt.legend()
+save_fig("gdregression")
+plt.show()
+
@@ -212,6 +249,8 @@ It is now the algorithm which wins essentially all ML competitions!!!
import matplotlib.pyplot as plt
import numpy as np
-from sklearn.model_selection import train_test_split
-import xgboost as xgb
-from sklearn.preprocessing import StandardScaler
+from sklearn.model_selection import train_test_split
+from sklearn.datasets import load_breast_cancer
import scikitplot as skplt
-from sklearn.metrics import mean_squared_error
+from sklearn.ensemble import GradientBoostingClassifier
+from sklearn.model_selection import cross_validate
-n = 100
-maxdegree = 6
+# Load the data
+cancer = load_breast_cancer()
-# Make data set.
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-
-error = np.zeros(maxdegree)
-bias = np.zeros(maxdegree)
-variance = np.zeros(maxdegree)
-polydegree = np.zeros(maxdegree)
-X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+#now scale the data
+from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
-for degree in range(maxdegree):
- model = xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200)
+gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0)
+gd_clf.fit(X_train_scaled, y_train)
+#Cross validation
+accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
- model.fit(X_train_scaled,y_train)
- y_pred = model.predict(X_test_scaled)
- polydegree[degree] = degree
- error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
- bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
- variance[degree] = np.mean( np.var(y_pred) )
- print('Max depth:', degree)
- print('Error:', error[degree])
- print('Bias^2:', bias[degree])
- print('Var:', variance[degree])
- print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
-
-plt.xlim(1,maxdegree-1)
-plt.plot(polydegree, error, label='Error')
-plt.plot(polydegree, bias, label='bias')
-plt.plot(polydegree, variance, label='Variance')
-plt.legend()
+import scikitplot as skplt
+y_pred = gd_clf.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+save_fig("gdclassiffierconfusion")
+plt.show()
+y_probas = gd_clf.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test, y_probas)
+save_fig("gdclassiffierroc")
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test, y_probas)
+save_fig("gdclassiffiercgain")
plt.show()
@@ -245,6 +242,8 @@ plt.show()
-As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now. +XGBoost or Extreme Gradient +Boosting, is an optimized distributed gradient boosting library +designed to be highly efficient, flexible and portable. It implements +machine learning algorithms under the Gradient Boosting +framework. XGBoost provides a parallel tree boosting that solve many +data science problems in a fast and accurate way. See the article by Chen and Guestrin. +
+The authors design and build a highly scalable end-to-end tree +boosting system. It has a theoretically justified weighted quantile +sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning. - -
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.datasets import load_breast_cancer
-from sklearn.preprocessing import LabelEncoder
-from sklearn.model_selection import cross_validate
-import scikitplot as skplt
-import xgboost as xgb
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-xg_clf = xgb.XGBClassifier()
-xg_clf.fit(X_train_scaled,y_train)
-
-y_test = xg_clf.predict(X_test_scaled)
-
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
-
-import scikitplot as skplt
-y_pred = xg_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-save_fig("xdclassiffierconfusion")
-plt.show()
-y_probas = xg_clf.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-save_fig("xdclassiffierroc")
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-save_fig("gdclassiffiercgain")
-plt.show()
-
-
-xgb.plot_tree(xg_clf,num_trees=0)
-plt.rcParams['figure.figsize'] = [50, 10]
-save_fig("xgtree")
-plt.show()
-
-xgb.plot_importance(xg_clf)
-plt.rcParams['figure.figsize'] = [5, 5]
-save_fig("xgparams")
-plt.show()
-+It is now the algorithm which wins essentially all ML competitions!!! +
diff --git a/doc/pub/week45/html/week45-bs.html b/doc/pub/week45/html/week45-bs.html index 716937c89..39586fc8c 100644 --- a/doc/pub/week45/html/week45-bs.html +++ b/doc/pub/week45/html/week45-bs.html @@ -41,64 +41,66 @@ Automatically generated HTML file from DocOnce source @@ -136,31 +138,33 @@ MathJax.Hub.Config({
-
@@ -219,7 +223,7 @@ MathJax.Hub.Config({
-
@@ -159,7 +159,28 @@ MathJax.Hub.Config({
+
+Geron's chapter 7. See also lecture from STK-IN4300, lecture 7. Chapter 9.2 of Hastie et al contains also a good discussion.
+
+Bagging, voting and random forests.
+
Random forests provide an improvement over bagged trees by way of a
@@ -205,7 +226,7 @@ this setting.
@@ -236,7 +257,7 @@ We will grow of forest of say \( B \) trees.
@@ -252,20 +273,20 @@ We will grow of forest of say \( B \) trees.
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
+print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
# Support vector machine
svm = SVC(gamma='auto', C=100)
svm.fit(X_train, y_train)
-print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
+print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
# Decision Trees
-deep_tree_clf = DecisionTreeClassifier(max_depth=None)
+deep_tree_clf = DecisionTreeClassifier(max_depth=None)
deep_tree_clf.fit(X_train, y_train)
-print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
+print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -274,13 +295,13 @@ X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Logistic Regression
logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Support Vector Machine
svm.fit(X_train_scaled, y_train)
-print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Decision Trees
deep_tree_clf.fit(X_train_scaled, y_train)
-print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
+print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
from sklearn.ensemble import RandomForestClassifier
@@ -292,13 +313,13 @@ Random_Forest_model = RandomForestClassifier(n_estimators=#Cross validation
accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = Random_Forest_model.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = Random_Forest_model.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
@@ -310,13 +331,13 @@ plt.show()
@@ -333,7 +354,7 @@ np.sum(y_pred == y_pred_rf) / len(y_pred)
The basic idea is to combine weak classifiers in order to create a good
@@ -350,7 +371,7 @@ them with a factor.
Boosting is a way of fitting an additive expansion in a set of
@@ -410,7 +431,7 @@ In iterative fitting or additive modeling, we minimize the cost function with re
The way we proceed is as follows (here we specialize to the squared-error cost function)
@@ -436,7 +457,7 @@ at the internal nodes, and the predictions at the terminal nodes.
To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
@@ -494,7 +515,7 @@ The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma
Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
@@ -535,7 +556,7 @@ $$
In our iterative procedure we define thus
@@ -569,7 +590,7 @@ where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
First, for any \( \beta > 0 \), we optimize \( G \) by setting
@@ -627,7 +648,7 @@ $$
The algorithm here is rather straightforward. Assume that our weak
@@ -651,7 +672,7 @@ where the function \( I() \) is one if we misclassify and zero if we classify co
With the above definitions we are now ready to set up the algorithm for AdaBoost.
@@ -692,7 +713,7 @@ observations that are missed in the previous iterations.
Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here.
@@ -714,7 +735,7 @@ ada_clf = AdaBoostClassifier(
algorithm="SAMME.R", learning_rate=0.5, random_state=42)
ada_clf.fit(X_train_scaled, y_train)
y_pred = ada_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = ada_clf.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
@@ -726,7 +747,7 @@ plt.show()
Here we present Drucker's AdaBoost tailored for regression.
@@ -755,7 +776,7 @@ Start by selecting a set of training data \( n \) and assign to each entry a wei
Gradient boosting is again a similar technique to Adaptive boosting,
@@ -770,7 +791,7 @@ function was the least squares function.
We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize
@@ -813,7 +834,7 @@ $$
Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that
@@ -835,7 +856,7 @@ and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \(
Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function
@@ -863,7 +884,7 @@ The way we proceed in an iterative fashion is to
We discuss here the difference between the steepest descent approach and gradient boosting by repeating our simple regression example above.
@@ -871,7 +892,7 @@ We discuss here the difference between the steepest descent approach and gradien
@@ -908,11 +929,11 @@ X_test_scaled = scaler.transform(X_test)
error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
variance[degree] = np.mean( np.var(y_pred) )
- print('Max depth:', degree)
- print('Error:', error[degree])
- print('Bias^2:', bias[degree])
- print('Var:', variance[degree])
- print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+ print('Max depth:', degree)
+ print('Error:', error[degree])
+ print('Bias^2:', bias[degree])
+ print('Var:', variance[degree])
+ print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
@@ -926,7 +947,7 @@ plt.show()
@@ -942,8 +963,8 @@ plt.show()
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -955,12 +976,12 @@ gd_clf = GradientBoostingClassifier(max_depth=3#Cross validation
accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = gd_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("gdclassiffierconfusion")
plt.show()
y_probas = gd_clf.predict_proba(X_test_scaled)
@@ -975,7 +996,7 @@ plt.show()
XGBoost or Extreme Gradient
@@ -996,7 +1017,7 @@ It is now the algorithm which wins essentially all ML competitions!!!
@@ -1035,11 +1056,11 @@ X_test_scaled = scaler.transform(X_test)
error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
variance[degree] = np.mean( np.var(y_pred) )
- print('Max depth:', degree)
- print('Error:', error[degree])
- print('Bias^2:', bias[degree])
- print('Var:', variance[degree])
- print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+ print('Max depth:', degree)
+ print('Error:', error[degree])
+ print('Bias^2:', bias[degree])
+ print('Var:', variance[degree])
+ print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
@@ -1052,7 +1073,7 @@ plt.show()
As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.
@@ -1071,8 +1092,8 @@ As you will see from the confusion matrix below, XGBoots does an excellent job o
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -1085,11 +1106,11 @@ xg_clf.fit(X_train_scaled,y_train)
y_test = xg_clf.predict(X_test_scaled)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = xg_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("xdclassiffierconfusion")
plt.show()
y_probas = xg_clf.predict_proba(X_test_scaled)
diff --git a/doc/pub/week45/html/week45-solarized.html b/doc/pub/week45/html/week45-solarized.html
index d1277cffd..43e4bfa95 100644
--- a/doc/pub/week45/html/week45-solarized.html
+++ b/doc/pub/week45/html/week45-solarized.html
@@ -35,64 +35,66 @@ div { text-align: justify; text-justify: inter-word; }
-
+
+Bagging, voting and random forests.
+
+
+
Random forests provide an improvement over bagged trees by way of a
@@ -183,7 +205,7 @@ this setting.
@@ -209,7 +231,7 @@ We will grow of forest of say \( B \) trees.
@@ -225,20 +247,20 @@ We will grow of forest of say \( B \) trees.
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
+print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
# Support vector machine
svm = SVC(gamma='auto', C=100)
svm.fit(X_train, y_train)
-print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
+print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
# Decision Trees
-deep_tree_clf = DecisionTreeClassifier(max_depth=None)
+deep_tree_clf = DecisionTreeClassifier(max_depth=None)
deep_tree_clf.fit(X_train, y_train)
-print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
+print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -247,13 +269,13 @@ X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Logistic Regression
logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Support Vector Machine
svm.fit(X_train_scaled, y_train)
-print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Decision Trees
deep_tree_clf.fit(X_train_scaled, y_train)
-print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
+print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
from sklearn.ensemble import RandomForestClassifier
@@ -265,13 +287,13 @@ Random_Forest_model = RandomForestClassifier(n_estimators=#Cross validation
accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = Random_Forest_model.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = Random_Forest_model.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
@@ -282,13 +304,13 @@ plt.show()
@@ -304,7 +326,7 @@ np.sum(y_pred == y_pred_rf) / len(y_pred)
The basic idea is to combine weak classifiers in order to create a good
@@ -321,7 +343,7 @@ them with a factor.
Boosting is a way of fitting an additive expansion in a set of
@@ -373,7 +395,7 @@ In iterative fitting or additive modeling, we minimize the cost function with re
The way we proceed is as follows (here we specialize to the squared-error cost function)
@@ -398,7 +420,7 @@ at the internal nodes, and the predictions at the terminal nodes.
To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
@@ -446,7 +468,7 @@ The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma
Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
@@ -481,7 +503,7 @@ $$
In our iterative procedure we define thus
@@ -509,7 +531,7 @@ where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
First, for any \( \beta > 0 \), we optimize \( G \) by setting
@@ -553,7 +575,7 @@ $$
The algorithm here is rather straightforward. Assume that our weak
@@ -575,7 +597,7 @@ where the function \( I() \) is one if we misclassify and zero if we classify co
With the above definitions we are now ready to set up the algorithm for AdaBoost.
@@ -615,7 +637,7 @@ observations that are missed in the previous iterations.
Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here.
@@ -637,7 +659,7 @@ ada_clf = AdaBoostClassifier(
algorithm="SAMME.R", learning_rate=0.5, random_state=42)
ada_clf.fit(X_train_scaled, y_train)
y_pred = ada_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = ada_clf.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
@@ -648,7 +670,7 @@ plt.show()
Here we present Drucker's AdaBoost tailored for regression.
@@ -676,7 +698,7 @@ Start by selecting a set of training data \( n \) and assign to each entry a wei
Gradient boosting is again a similar technique to Adaptive boosting,
@@ -691,7 +713,7 @@ function was the least squares function.
We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize
@@ -726,7 +748,7 @@ $$
Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that
@@ -744,7 +766,7 @@ and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \(
Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function
@@ -770,7 +792,7 @@ The way we proceed in an iterative fashion is to
We discuss here the difference between the steepest descent approach and gradient boosting by repeating our simple regression example above.
@@ -778,7 +800,7 @@ We discuss here the difference between the steepest descent approach and gradien
@@ -815,11 +837,11 @@ X_test_scaled = scaler.transform(X_test)
error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
variance[degree] = np.mean( np.var(y_pred) )
- print('Max depth:', degree)
- print('Error:', error[degree])
- print('Bias^2:', bias[degree])
- print('Var:', variance[degree])
- print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+ print('Max depth:', degree)
+ print('Error:', error[degree])
+ print('Bias^2:', bias[degree])
+ print('Var:', variance[degree])
+ print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
@@ -832,7 +854,7 @@ plt.show()
@@ -848,8 +870,8 @@ plt.show()
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -861,12 +883,12 @@ gd_clf = GradientBoostingClassifier(max_depth=3#Cross validation
accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = gd_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("gdclassiffierconfusion")
plt.show()
y_probas = gd_clf.predict_proba(X_test_scaled)
@@ -880,7 +902,7 @@ plt.show()
XGBoost or Extreme Gradient
@@ -901,7 +923,7 @@ It is now the algorithm which wins essentially all ML competitions!!!
@@ -940,11 +962,11 @@ X_test_scaled = scaler.transform(X_test)
error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
variance[degree] = np.mean( np.var(y_pred) )
- print('Max depth:', degree)
- print('Error:', error[degree])
- print('Bias^2:', bias[degree])
- print('Var:', variance[degree])
- print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+ print('Max depth:', degree)
+ print('Error:', error[degree])
+ print('Bias^2:', bias[degree])
+ print('Var:', variance[degree])
+ print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
@@ -956,7 +978,7 @@ plt.show()
As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.
@@ -975,8 +997,8 @@ As you will see from the confusion matrix below, XGBoots does an excellent job o
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -989,11 +1011,11 @@ xg_clf.fit(X_train_scaled,y_train)
y_test = xg_clf.predict(X_test_scaled)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = xg_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("xdclassiffierconfusion")
plt.show()
y_probas = xg_clf.predict_proba(X_test_scaled)
diff --git a/doc/pub/week45/html/week45.html b/doc/pub/week45/html/week45.html
index a3859ad05..7617607e2 100644
--- a/doc/pub/week45/html/week45.html
+++ b/doc/pub/week45/html/week45.html
@@ -40,64 +40,66 @@ div { text-align: justify; text-justify: inter-word; }
-
+
+Bagging, voting and random forests.
+
+
+
Random forests provide an improvement over bagged trees by way of a
@@ -188,7 +210,7 @@ this setting.
@@ -214,7 +236,7 @@ We will grow of forest of say \( B \) trees.
@@ -230,20 +252,20 @@ We will grow of forest of say \( B \) trees.
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
# Logistic Regression
logreg = LogisticRegression(solver='lbfgs')
logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
+print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
# Support vector machine
svm = SVC(gamma='auto', C=100)
svm.fit(X_train, y_train)
-print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
+print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
# Decision Trees
-deep_tree_clf = DecisionTreeClassifier(max_depth=None)
+deep_tree_clf = DecisionTreeClassifier(max_depth=None)
deep_tree_clf.fit(X_train, y_train)
-print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
+print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -252,13 +274,13 @@ X_train_scaled = scaler= scaler.transform(X_test)
# Logistic Regression
logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Support Vector Machine
svm.fit(X_train_scaled, y_train)
-print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
# Decision Trees
deep_tree_clf.fit(X_train_scaled, y_train)
-print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
+print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
from sklearn.ensemble import RandomForestClassifier
@@ -270,13 +292,13 @@ Random_Forest_model = RandomForestClassifier
Random_Forest_model.fit(X_train_scaled, y_train)
#Cross validation
accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = Random_Forest_model.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = Random_Forest_model.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
@@ -287,13 +309,13 @@ plt.show()
@@ -309,7 +331,7 @@ np.sum(y_pred =
The basic idea is to combine weak classifiers in order to create a good
@@ -326,7 +348,7 @@ them with a factor.
Boosting is a way of fitting an additive expansion in a set of
@@ -378,7 +400,7 @@ In iterative fitting or additive modeling, we minimize the cost function with re
The way we proceed is as follows (here we specialize to the squared-error cost function)
@@ -403,7 +425,7 @@ at the internal nodes, and the predictions at the terminal nodes.
To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
@@ -451,7 +473,7 @@ The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma
Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
@@ -486,7 +508,7 @@ $$
In our iterative procedure we define thus
@@ -514,7 +536,7 @@ where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
First, for any \( \beta > 0 \), we optimize \( G \) by setting
@@ -558,7 +580,7 @@ $$
The algorithm here is rather straightforward. Assume that our weak
@@ -580,7 +602,7 @@ where the function \( I() \) is one if we misclassify and zero if we classify co
With the above definitions we are now ready to set up the algorithm for AdaBoost.
@@ -620,7 +642,7 @@ observations that are missed in the previous iterations.
Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here.
@@ -642,7 +664,7 @@ ada_clf = AdaBoostClassifier(
algorithm="SAMME.R", learning_rate=0.5, random_state=42)
ada_clf.fit(X_train_scaled, y_train)
y_pred = ada_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
plt.show()
y_probas = ada_clf.predict_proba(X_test_scaled)
skplt.metrics.plot_roc(y_test, y_probas)
@@ -653,7 +675,7 @@ plt.show()
Here we present Drucker's AdaBoost tailored for regression.
@@ -681,7 +703,7 @@ Start by selecting a set of training data \( n \) and assign to each entry a wei
Gradient boosting is again a similar technique to Adaptive boosting,
@@ -696,7 +718,7 @@ function was the least squares function.
We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize
@@ -731,7 +753,7 @@ $$
Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that
@@ -749,7 +771,7 @@ and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \(
Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function
@@ -775,7 +797,7 @@ The way we proceed in an iterative fashion is to
We discuss here the difference between the steepest descent approach and gradient boosting by repeating our simple regression example above.
@@ -783,7 +805,7 @@ We discuss here the difference between the steepest descent approach and gradien
@@ -820,11 +842,11 @@ X_test_scaled = scaler= np.mean( np.mean((y_test - y_pred)**2) )
bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
variance[degree] = np.mean( np.var(y_pred) )
- print('Max depth:', degree)
- print('Error:', error[degree])
- print('Bias^2:', bias[degree])
- print('Var:', variance[degree])
- print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+ print('Max depth:', degree)
+ print('Error:', error[degree])
+ print('Bias^2:', bias[degree])
+ print('Var:', variance[degree])
+ print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
@@ -837,7 +859,7 @@ plt.show()
@@ -853,8 +875,8 @@ plt.show()
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -866,12 +888,12 @@ gd_clf = GradientBoostingClassifier(max_dept
gd_clf.fit(X_train_scaled, y_train)
#Cross validation
accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = gd_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("gdclassiffierconfusion")
plt.show()
y_probas = gd_clf.predict_proba(X_test_scaled)
@@ -885,7 +907,7 @@ plt.show()
XGBoost or Extreme Gradient
@@ -906,7 +928,7 @@ It is now the algorithm which wins essentially all ML competitions!!!
@@ -945,11 +967,11 @@ X_test_scaled = scaler= np.mean( np.mean((y_test - y_pred)**2) )
bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
variance[degree] = np.mean( np.var(y_pred) )
- print('Max depth:', degree)
- print('Error:', error[degree])
- print('Bias^2:', bias[degree])
- print('Var:', variance[degree])
- print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+ print('Max depth:', degree)
+ print('Error:', error[degree])
+ print('Bias^2:', bias[degree])
+ print('Var:', variance[degree])
+ print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
plt.xlim(1,maxdegree-1)
plt.plot(polydegree, error, label='Error')
@@ -961,7 +983,7 @@ plt.show()
As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.
@@ -980,8 +1002,8 @@ As you will see from the confusion matrix below, XGBoots does an excellent job o
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
+print(X_train.shape)
+print(X_test.shape)
#now scale the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
@@ -994,11 +1016,11 @@ xg_clf.fit(X_train_scaled,y_train)
y_test = xg_clf.predict(X_test_scaled)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
import scikitplot as skplt
y_pred = xg_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
save_fig("xdclassiffierconfusion")
plt.show()
y_probas = xg_clf.predict_proba(X_test_scaled)
diff --git a/doc/pub/week45/ipynb/ipynb-week45-src.tar.gz b/doc/pub/week45/ipynb/ipynb-week45-src.tar.gz
index f30ba752f..f8420afba 100644
Binary files a/doc/pub/week45/ipynb/ipynb-week45-src.tar.gz and b/doc/pub/week45/ipynb/ipynb-week45-src.tar.gz differ
diff --git a/doc/pub/week45/ipynb/week45.ipynb b/doc/pub/week45/ipynb/week45.ipynb
index 80433a3e0..d7b453b7b 100644
--- a/doc/pub/week45/ipynb/week45.ipynb
+++ b/doc/pub/week45/ipynb/week45.ipynb
@@ -10,12 +10,24 @@
" \n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
"\n",
- "Date: **Sep 16, 2020**\n",
+ "Date: **Oct 31, 2020**\n",
"\n",
"Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
"\n",
"\n",
+ "## Overview of week 45\n",
+ "\n",
+ "* \"Thursday: Wrapping up from last week. Bagging and Random forests.\n",
+ "\n",
+ "* \"Friday: Boosting and gradient boosting\n",
+ "\n",
+ "Geron's chapter 7. See also lecture from [STK-IN4300, lecture 7](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_7.pdf). Chapter 9.2 of Hastie et al contains also a good discussion.\n",
+ "\n",
+ "\n",
+ "## Thursday\n",
+ "\n",
+ "Bagging, voting and random forests.\n",
"\n",
"\n",
"## Random forests\n",
@@ -1208,5 +1220,5 @@
],
"metadata": {},
"nbformat": 4,
- "nbformat_minor": 2
+ "nbformat_minor": 4
}
diff --git a/doc/src/week45/week45.do.txt b/doc/src/week45/week45.do.txt
index 13c6037d4..a7554dc16 100644
--- a/doc/src/week45/week45.do.txt
+++ b/doc/src/week45/week45.do.txt
@@ -2,6 +2,20 @@ TITLE: Week 45: Random Forests and Boosting
AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
DATE: today
+!split
+===== Overview of week 45 =====
+
+* "Thursday: Wrapping up from last week. Bagging and Random forests.
+* "Friday: Boosting and gradient boosting
+
+
+Geron's chapter 7. See also lecture from "STK-IN4300, lecture 7":"https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_7.pdf". Chapter 9.2 of Hastie et al contains also a good discussion.
+
+
+!split
+===== Thursday =====
+
+Bagging, voting and random forests.
!split
Random forests
+Overview of week 45
+
+
+
+Thursday
+
+Random forests
Random Forest Algorithm
+Random Forest Algorithm
The algorithm described here can be applied to both classification and regression problems.
Random Forests Compared with other Methods on the Cancer Data
+Random Forests Compared with other Methods on the Cancer Data
Compare Bagging on Trees with Random Forests
+Compare Bagging on Trees with Random Forests
bag_clf = BaggingClassifier(
DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
- n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
+ n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
Boosting, a Bird's Eye View
+Boosting, a Bird's Eye View
What is boosting? Additive Modelling/Iterative Fitting
+What is boosting? Additive Modelling/Iterative Fitting
Iterative Fitting, Regression and Squared-error Cost Function
+Iterative Fitting, Regression and Squared-error Cost Function
Squared-Error Example and Iterative Fitting
+Squared-Error Example and Iterative Fitting
Iterative Fitting, Classification and AdaBoost
+Iterative Fitting, Classification and AdaBoost
Adaptive Boosting, AdaBoost
+Adaptive Boosting, AdaBoost
Building up AdaBoost
+Building up AdaBoost
Adaptive boosting: AdaBoost, Basic Algorithm
+Adaptive boosting: AdaBoost, Basic Algorithm
Basic Steps of AdaBoost
+Basic Steps of AdaBoost
AdaBoost Examples
+AdaBoost Examples
AdaBoost for Regression
+AdaBoost for Regression
Gradient boosting: Basics with Steepest Descent
+Gradient boosting: Basics with Steepest Descent
The Squared-Error again! Steepest Descent
+The Squared-Error again! Steepest Descent
Steepest Descent Example
+Steepest Descent Example
Gradient Boosting, algorithm
+Gradient Boosting, algorithm
Gradient Boosting Example, Regression
+Gradient Boosting Example, Regression
Gradient Boosting, Examples of Regression
+Gradient Boosting, Examples of Regression
Gradient Boosting, Classification Example
+Gradient Boosting, Classification Example
XGBoost: Extreme Gradient Boosting
+XGBoost: Extreme Gradient Boosting
Regression Case
+Regression Case
Xgboost on the Cancer Data
+Xgboost on the Cancer Data
Sep 16, 2020
Oct 31, 2020
-Random forests
+Overview of week 45
+
+
+
+
+Geron's chapter 7. See also lecture from STK-IN4300, lecture 7. Chapter 9.2 of Hastie et al contains also a good discussion.
+
+
+
+Thursday
+
+
+
+Random forests
-Random Forest Algorithm
+Random Forest Algorithm
The algorithm described here can be applied to both classification and regression problems.
-Random Forests Compared with other Methods on the Cancer Data
+Random Forests Compared with other Methods on the Cancer Data
-Compare Bagging on Trees with Random Forests
+Compare Bagging on Trees with Random Forests
bag_clf = BaggingClassifier(
DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
- n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
+ n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
-Boosting, a Bird's Eye View
+Boosting, a Bird's Eye View
-What is boosting? Additive Modelling/Iterative Fitting
+What is boosting? Additive Modelling/Iterative Fitting
-Iterative Fitting, Regression and Squared-error Cost Function
+Iterative Fitting, Regression and Squared-error Cost Function
-Squared-Error Example and Iterative Fitting
+Squared-Error Example and Iterative Fitting
-Iterative Fitting, Classification and AdaBoost
+Iterative Fitting, Classification and AdaBoost
-Adaptive Boosting, AdaBoost
+Adaptive Boosting, AdaBoost
-Building up AdaBoost
+Building up AdaBoost
-Adaptive boosting: AdaBoost, Basic Algorithm
+Adaptive boosting: AdaBoost, Basic Algorithm
-Basic Steps of AdaBoost
+Basic Steps of AdaBoost
-AdaBoost Examples
+AdaBoost Examples
-AdaBoost for Regression
+AdaBoost for Regression
-Gradient boosting: Basics with Steepest Descent
+Gradient boosting: Basics with Steepest Descent
-The Squared-Error again! Steepest Descent
+The Squared-Error again! Steepest Descent
-Steepest Descent Example
+Steepest Descent Example
-Gradient Boosting, algorithm
+Gradient Boosting, algorithm
-Gradient Boosting Example, Regression
+Gradient Boosting Example, Regression
-Gradient Boosting, Examples of Regression
+Gradient Boosting, Examples of Regression
-Gradient Boosting, Classification Example
+Gradient Boosting, Classification Example
-XGBoost: Extreme Gradient Boosting
+XGBoost: Extreme Gradient Boosting
-Regression Case
+Regression Case
-Xgboost on the Cancer Data
+Xgboost on the Cancer Data
Sep 16, 2020
Oct 31, 2020
-Random forests
+Overview of week 45
+
+
+
+
+Geron's chapter 7. See also lecture from STK-IN4300, lecture 7. Chapter 9.2 of Hastie et al contains also a good discussion.
+
+
+
+Thursday
+
+
+
+Random forests
-Random Forest Algorithm
+Random Forest Algorithm
The algorithm described here can be applied to both classification and regression problems.
-Random Forests Compared with other Methods on the Cancer Data
+Random Forests Compared with other Methods on the Cancer Data
-Compare Bagging on Trees with Random Forests
+Compare Bagging on Trees with Random Forests
bag_clf = BaggingClassifier(
DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
- n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
+ n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
-Boosting, a Bird's Eye View
+Boosting, a Bird's Eye View
-What is boosting? Additive Modelling/Iterative Fitting
+What is boosting? Additive Modelling/Iterative Fitting
-Iterative Fitting, Regression and Squared-error Cost Function
+Iterative Fitting, Regression and Squared-error Cost Function
-Squared-Error Example and Iterative Fitting
+Squared-Error Example and Iterative Fitting
-Iterative Fitting, Classification and AdaBoost
+Iterative Fitting, Classification and AdaBoost
-Adaptive Boosting, AdaBoost
+Adaptive Boosting, AdaBoost
-Building up AdaBoost
+Building up AdaBoost
-Adaptive boosting: AdaBoost, Basic Algorithm
+Adaptive boosting: AdaBoost, Basic Algorithm
-Basic Steps of AdaBoost
+Basic Steps of AdaBoost
-AdaBoost Examples
+AdaBoost Examples
-AdaBoost for Regression
+AdaBoost for Regression
-Gradient boosting: Basics with Steepest Descent
+Gradient boosting: Basics with Steepest Descent
-The Squared-Error again! Steepest Descent
+The Squared-Error again! Steepest Descent
-Steepest Descent Example
+Steepest Descent Example
-Gradient Boosting, algorithm
+Gradient Boosting, algorithm
-Gradient Boosting Example, Regression
+Gradient Boosting Example, Regression
-Gradient Boosting, Examples of Regression
+Gradient Boosting, Examples of Regression
-Gradient Boosting, Classification Example
+Gradient Boosting, Classification Example
-XGBoost: Extreme Gradient Boosting
+XGBoost: Extreme Gradient Boosting
-Regression Case
+Regression Case
-Xgboost on the Cancer Data
+Xgboost on the Cancer Data